A072962 - OEIS

%I #13 Nov 09 2024 06:38:58

%S 1,20,1071,107104,17201225,4053135456,1318104508735,565989104282624,

%T 310299479406324369,211554189796535488000,175592153482084893991151,

%U 174356954302176729972264960,204111110614488911169799727641,278218647289052493421682954399744

%N Unsigned reduced Euler characteristic for the matroid complex of cycle matroid for complete bipartite graph K_{n,n}.

%C We will denote this number by a(n,n). It is also the value of the Tutte polynomial T_{G}(0,1) for G=K_{n,n}.

%C The formula given for a(s,t) is valid for all s>1 and t>0. Also note that a(s,t) = a(t,s) because K_{s,t} = K_{t,s}. For small values of s we have the following formulas: a(2,t)=t-1, a(3,t)=2^{t-2}(t-1)(3t-4), a(4,t)=3^{t-3}(t-1)(16t^2-41t+27), a(5,t)=4^{t-4}(t-1)(125t^3-376t^2+378t-133)

%D I. Novik, A. Postnikov and B. Sturmfels: Syzygies of oriented matroids, Duke Math. J. 111 (2002), no. 2, 287-317.

%H Woong Kook and Kang-Ju Lee, <a href="https://doi.org/10.1016/j.ejc.2018.04.001">Möbius coinvariants and bipartite edge-rooted forests</a>, European Journal of Combinatorics, Volume 71, June 2018, Pages 180-193.

%H I. Novik, A. Postnikov and B. Sturmfels, <a href="https://arxiv.org/abs/math/0009241">Syzygies of oriented matroids</a>, arXiv:math/0009241 [math.CO], 2000.

%F a(n) = a(n, n) where a(s, t) = Sum_{i=0..s-2} (-1)^i * binomial(s-1,i) * w(s-1-i, t), where s,t>1 and an e.g.f. for w(a, b) is given by exp( Sum_{i,j>0} i^(j-1) * j^(i-1) * (j-1) * x^i * y^j / (i! * j!) ).

%e a(2,2)=1. Since K_{2,2} is a cycle with four edges, the matroid complex of cycle matroid for K_{2,2} is the 2-skeleton of standard 3-simplex. Therefore the unsigned reduced Euler characteristic for this complex is |-1+4-6+4|=1

%Y Cf. A057817.

%K nonn

%O 2,2

%A W. Kook and L. Thoma (andrewk(AT)math.uri.edu), Aug 20 2002

%E More terms from _Sean A. Irvine_, Nov 08 2024